Parallelogram : A parallelogram is a quadrilateral, in which both the pairs of opposite sides are parallel. The quadrilateral ABCD, drawn alongside, is a parallelogram; since, AB is parallel to DC and AD is parallel to BC i.e.

AB || DC and AD || BC. Also, in a parallelogram ABCD: (i) opposite sides are equal: i.e. AB = DC and AD = BC. (ii) opposite angles are equal: i.e. ∠ABC = ∠ADC and ∠BCD = ∠BAD (iii) diagonals bisect each other : i.e. OA = OC = 12 AC and OB = OD = 12 BD.

Some special types of Parallelograms (a) Rhombus : A rhombus is a parallelogram in which all its sides are equal.

∴In a rhombus ABCD : (i) opposite sides are parallel: i.e. AB||DC and AD||BC. (ii) all the sides are equal: i.e. AB = BC = CD = DA. (iii) opposite angles are equal: i.e. ∠A = ∠C and ∠B = ∠D. (iv) diagonals bisect each other at right angle : i.e. OA= OC = 12 AC ; OB = OD = 12 BD. and ∠AOB= ∠BOC = ∠COD = ∠AOD = 90° (v) diagonals bisect the angles at the vertices : i.e. ∠1 =∠2 ;∠3 = ∠4 ; ∠5 = ∠6 and ∠7 =∠8.

(b) Rectangle : A rectangle is a parallelogram whose any angle is 90°. A rectangle is also defined as a quadrilateral whose each angle is 90°.

Note : If any angle of a parallelogram is 90° ; automatically its each angle is 90° ; the reason being that the opposite angles of a parallelogram are equal. Also, in a rectangle: (i) opposite sides are parallel. (ii) opposite sides are equal. (iii) each angle is 90°. (iv) diagonals are equal. (v) diagonals bisect each other.

(c) Square : A square is a parallelogram, whose all side are equal and each angle is 90°.

A square can also be defined as : (i) a rhombus whose any angle is 90°. (ii) a rectangle whose all sides are equal. (iii) a quadrilateral whose all sides are equal and each angle is 90°. ∴ If ABCD is a square :

(i) all its sides are equal, i.e. AB = BC = CD = DA (ii) each angle of it is 90°. i.e. ∠A = ∠B = ∠C = ∠D = 90°. Also, (iii) diagonals are equal. i.e. AC = BD. (iv) diagonals bisect each other at 90°. i.e. OA = OC =12 AC;OB = OD = 12 BD and ∠AOB = ∠BOC = ∠COD = ∠DOA = 90°. Since, diagonals AC and BD are equal; therefore ; OA = OC = OB = OD. (v) diagonals bisect the angles at the vertices i.e. ∠1 = ∠2 = 45° [∵ ∠1 + ∠2 = 90°] Similarly; ∠3 = ∠4 = 45° ; ∠5 – ∠6 = 45° and ∠7 = ∠8 = 45°.